This would connect it to the PIE verbal root *ped- "to walk," from *ped- (1) "foot" (see foot (n.)). With widespread sense development: to "reach," "deliver," "effect," "make (butter), churn" (19c.), "restore to consciousness" (1620s), also various nautical senses from 16c.-17c.; meaning "to bring in as equivalent or price" is from c. 1600. In 17c. writers on language didn't derive a word's etymology; they fetched it. As what a dog does, c. 1600, originally fetch-and-carry. Variant form fet, a derivation of the original Old English version of the word, survived as a competitor until 17c. Related: Fetched; fetching.

"apparition of a living person, specter, a double," 1787, an English dialect word of unknown origin (see OED for discussion).

A peculiarly weird type of apparition is the wraith (q.v.) or double, of which the Irish fetch is a variant. The wraith is an exact facsimile of a living person, who may himself see it. Goethe, Shelley, and other famous men are said to have seen their own wraiths. The fetch makes its appearance shortly before the death of the person it represents, either to himself or his friends, or both. [Lewis Spence, "An Encyclopedia of Occultism," 1920]